### Logarithmic functions and their graphs

Question Sample Titled 'Logarithmic functions and their graphs'

A function in the form ${f{{\left({x}\right)}}}={{\log}_{{a}}{x}}$ , where ${a}\gt{0}$ and ${a}\ne{1}$ , is called a logarithmic function.
Range of ${a}$${a}\gt{1}$${0}\lt{a}\lt{1}$
Graph of ${y}={{\log}_{{a}}{x}}$
Common characteristicsThe graph cuts the x-axis at ${\left({1},{0}\right)}$. The graph lies on the right-hand side of the y-axis. The graph has neither a maximum point, a minimum point nor an axis of symmetry.
DifferencesFor ${0}\lt{x}\lt{1}$, ${{\log}_{{a}}{x}}\lt{0}$ . For ${x}\gt{1}$, ${{\log}_{{a}}{x}}\gt{0}$. The value of ${{\log}_{{a}}{x}}$ increases as ${x}$ increases. As ${x}$ increases, the rate of increase of ${{\log}_{{a}}{x}}$decreases. For ${0}\lt{x}\lt{1}$, ${{\log}_{{a}}{x}}\gt{0}$. For ${x}\gt{1}$, ${{\log}_{{a}}{x}}\lt{0}$. The value of ${{\log}_{{a}}{x}}$ decreases as ${x}$ increases. As ${x}$ increases, the rate of decrease of ${{\log}_{{a}}{x}}$ decreases.
Note: The graph of ${y}={{\log}_{{a}}{x}}$ and ${y}={{\log}_{{\tfrac{{1}}{{a}}}}{x}}$ have reflectional symmetry about the x-axis. The graph of ${y}={a}^{{x}}$ and ${y}={{\log}_{{a}}{x}}$ have reflectional symmetry about the line ${y}={x}$.

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